Subgroups — C₅×#22719 #11945439

Order 1280.

790 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₄×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic
3	C₅×#1950	C₂	1	2	yes	maximal, complemented
4	C₅×#1855	C₂	1	4	yes	maximal, complemented
5	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1811	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1738	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₅×#436	—	2	2	no	—
13	C₅×#436	C₄	1	4	yes	—
14	C₅×#436	C₄	1	4	yes	—
15	C₅×#435	—	2	4	no	—
16	C₅×C₈×D₄	—	2	2	no	—
17	C₂×C₅×#108	C₂²	1	2	yes	—
18	C₂×C₅×#108	C₂²	1	4	yes	—
19	C₅×#423	C₂²	1	2	yes	—
20	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₅×#407	C₂²	1	4	yes	—
22	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₄×C₅×M₁₆	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₅×#341	C₂²	1	1	yes	characteristic
26	C₅×#341	C₂²	1	1	yes	characteristic
27	C₅×#341	C₂²	1	2	yes	—
28	C₅×#341	C₂²	1	2	yes	—
29	C₅×#341	C₂²	1	4	yes	—
30	#22719	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic
32	C₂³×C₄×C₅	C₂³	1	2	yes	—
33	C₂×C₅×M₁₆	—	2	4	no	—
34	C₂×C₅×M₁₆	—	2	8	no	—
35	C₂×C₅×M₁₆	—	4	4	no	—
36	C₂×C₅×M₁₆	—	4	4	no	—
37	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
38	C₂×C₅×M₁₆	C₂³	1	2	yes	—
39	C₂×C₅×M₁₆	C₂³	1	4	yes	—
40	C₂×C₅×M₁₆	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	—	2	4	no	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
46	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
47	C₄×C₅×D₄	D₄	1	2	yes	—
48	C₄×C₅×D₄	D₄	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	Q₈	1	1	yes	characteristic
50	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×#108	—	2	4	no	—
53	C₅×#108	—	4	4	no	—
54	C₅×C₄⋊C₈	—	2	2	no	—
55	C₅×C₄⋊C₈	—	2	2	no	—
56	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
58	C₅×C₂²⋊C₈	—	2	4	no	—
59	C₅×C₂²⋊C₈	—	2	4	no	—
60	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
61	C₅×#97	—	2	2	no	—
62	C₄×C₅×C₈	—	2	2	no	—
63	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
64	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
65	D₄×M₁₆	C₂×C₅	1	2	yes	—
66	C₂×#436	C₂×C₅	1	1	yes	characteristic
67	#1950	C₂×C₅	1	2	yes	complemented
68	#1855	C₂×C₅	1	4	yes	complemented
69	#1837	C₂×C₅	1	1	yes	characteristic, complemented
70	#1811	C₂×C₅	1	2	yes	complemented
71	C₄×#108	C₂×C₅	1	1	yes	characteristic, complemented
72	#1738	C₂×C₅	1	2	yes	complemented
73	C₂⁴×C₅	C₂×Q₈	1	2	yes	—
74	C₂×C₅×D₄	—	4	2	no	—
75	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
76	C₂×C₅×D₄	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	—	2	4	no	—
87	C₅×M₁₆	—	2	2	no	—
88	C₅×M₁₆	—	2	4	no	—
89	C₅×M₁₆	—	2	2	no	—
90	C₅×M₁₆	—	4	8	no	—
91	C₅×M₁₆	—	4	8	no	—
92	C₅×M₁₆	#41	1	4	yes	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	4	8	no	—
100	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
103	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
104	C₅×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
105	C₅×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
106	C₅×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
107	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
111	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
112	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
113	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
114	#436	—	2	2	no	—
115	#436	C₄×C₅	1	4	yes	—
116	#436	C₄×C₅	1	4	yes	—
117	#435	—	2	4	no	—
118	C₈×D₄	—	2	2	no	—
119	C₂×#108	C₂²×C₅	1	2	yes	—
120	C₂×#108	C₂²×C₅	1	4	yes	—
121	#423	C₂²×C₅	1	2	yes	—
122	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
123	#407	C₂²×C₅	1	4	yes	—
124	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
125	C₄×M₁₆	C₂²×C₅	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
127	#341	C₂²×C₅	1	1	yes	characteristic
128	#341	C₂²×C₅	1	2	yes	—
129	#341	C₂²×C₅	1	2	yes	—
130	#341	C₂²×C₅	1	4	yes	—
131	#341	C₂²×C₅	1	1	yes	characteristic
132	C₂³×C₅	—	2	4	no	—
133	C₂³×C₅	—	4	2	no	—
134	C₂³×C₅	—	4	2	no	—
135	C₂³×C₅	C₂×C₄⋊C₄	1	4	yes	—
136	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
137	C₅×D₄	—	2	4	no	—
138	C₅×D₄	—	4	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
150	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×#41	1	2	yes	—
154	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	2	4	no	—
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	—	2	4	no	—
159	C₅×C₈	—	4	4	no	—
160	C₅×C₈	—	4	2	no	—
161	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₅	1	2	yes	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	8	no	—
165	C₂×M₁₆	—	4	4	no	—
166	C₂×M₁₆	—	4	4	no	—
167	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
168	C₂×M₁₆	C₂³×C₅	1	4	yes	—
169	C₂×M₁₆	C₂³×C₅	1	2	yes	—
170	C₂×M₁₆	C₂³×C₅	1	2	yes	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₅	1	2	yes	—
174	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
175	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
176	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
177	C₄×D₄	C₅×D₄	1	2	yes	—
178	C₄×D₄	C₅×D₄	1	2	yes	—
179	C₂×C₄⋊C₄	C₅×Q₈	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
181	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
182	#108	—	2	4	no	—
183	#108	—	4	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
187	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
188	C₂²⋊C₈	—	2	4	no	—
189	C₂²⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
191	#97	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
194	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	—	8	4	no	—
201	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
202	C₂²×C₅	#554	1	2	yes	—
203	C₄×C₅	—	2	4	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	#430	1	2	yes	—
208	C₄×C₅	C₄×#41	1	1	yes	characteristic
209	C₄×C₅	#518	1	1	yes	characteristic
210	C₄×C₅	#546	1	2	yes	—
211	C₂⁴	C₂×C₅×Q₈	1	2	yes	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
214	C₂×D₄	C₅×C₄⋊C₄	1	2	yes	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
218	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
219	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
221	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
222	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	8	no	—
229	M₁₆	—	4	8	no	—
230	M₁₆	C₅×#41	1	4	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	8	no	—
238	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
241	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
242	C₄⋊C₄	C₅×C₄⋊C₄	1	2	yes	—
243	C₂²⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
244	C₂²⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
245	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
247	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
248	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	4	4	no	—
251	C₂×C₅	#2349	1	1	yes	characteristic
252	C₂×C₅	C₂×#430	1	1	yes	characteristic
253	C₂×C₅	#2916	1	1	yes	characteristic
254	C₂³	—	2	4	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	C₂×C₅×C₄⋊C₄	1	4	yes	—
258	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
259	D₄	—	2	4	no	—
260	D₄	—	4	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
270	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
272	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₅×#41	1	2	yes	—
276	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	4	no	—
280	C₈	—	2	4	no	—
281	C₈	—	4	4	no	—
282	C₈	—	4	2	no	—
283	C₅	#22719	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	8	4	no	—
290	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
291	C₂²	C₅×#554	1	2	yes	—
292	C₄	—	2	4	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	C₅×#430	1	2	yes	—
297	C₄	C₄×C₅×#41	1	1	yes	characteristic
298	C₄	C₅×#518	1	1	yes	characteristic
299	C₄	C₅×#546	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	4	4	no	—
302	C₂	C₅×#2349	1	1	yes	characteristic
303	C₂	C₂×C₅×#430	1	1	yes	characteristic
304	C₂	C₅×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂²×C₅×D₄	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₅×#1950	C₂	semidirect
4. C₅×#1855	C₂	semidirect
5. C₅×#1837	C₂	semidirect
6. C₅×#1811	C₂	semidirect
7. C₄×C₅×#108	C₂	semidirect
8. C₅×#1738	C₂	semidirect
30. #22719	C₅	direct (N × H)
67. #1950	C₂×C₅	semidirect
68. #1855	C₂×C₅	semidirect
69. #1837	C₂×C₅	semidirect
70. #1811	C₂×C₅	semidirect
71. C₄×#108	C₂×C₅	semidirect
72. #1738	C₂×C₅	semidirect
283. C₅	#22719	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×D₄×M₁₆	2
C₂×C₅×#436	2
C₅×#1950	2
C₅×#1855	2
C₅×#1837	2
C₅×#1811	2
C₄×C₅×#108	2
C₅×#1738	2
#22719	5

Subgroups — C5×#22719 #11945439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22719 #11945439